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5
46
Functions of real variables
Modes of convergence
5a
Introduction
. . . . . . . . . . . . . . . . . . . . .
46
5b
Local convergence in measure . . . . . . . . . . .
47
5c
Convergence almost everywhere
. . . . . . . . .
49
5d
Dominated convergence; integrals depending on
a parameter . . . . . . . . . . . . . . . . . . . . . .
51
5e
One-sided bounds . . . . . . . . . . . . . . . . . .
53
5f
Spaces Lp . . . . . . . . . . . . . . . . . . . . . . .
56
.
5a
Introduction
If one has a sequence x1 , x2 , x3 , · · · ∈ R of real numbers xn , it
is unambiguous what it means for that sequence to converge to
a limit x ∈ R . . . If, however, one has a sequence f1 , f2 , f3 , . . .
of functions fn : X → R . . . and a putative limit f : X → R
. . . there can now be many different ways in which the sequence
fn may or may not converge to the limit f . . . Once X becomes
infinite, the functions fn acquire an infinite number of degrees of
freedom, and this allows them to approach f in any number of
inequivalent ways. . . . However, pointwise and uniform convergence are only two of dozens of many other modes of convergence
that are of importance in analysis.
Tao, Sect. 1.5 “Modes of convergence”.
When speaking about functions fn on a measure space we really mean, as
usual, equivalence classes [fn ]. Thus we restrict ourselves to such modes of
convergence that are insensitive to arbitrary change of fn on a null set. (Do
you bother whether the null set depends on n?) We replace the pointwise
convergence with the convergence almost everywhere,
fn → f a.e.
⇐⇒
µ{x : fn (x) 6→ f (x)} = 0 ⇐⇒
⇐⇒ fn → f pointwise outside some null set,
and the uniform convergence with the convergence uniform almost everywhere,
fn → f in L∞
⇐⇒
fn → f uniformly outside some null set
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Functions of real variables
47
(why “L∞ ”? wait for 5f7). On the other hand, the given measure leads to
new modes of convergence, such as
Z
fn → f in L1 ⇐⇒
|fn − f | dµ → 0 .
X
5b
Local convergence in measure
This is the weakest among all modes of convergence that we need. (That is,
every other convergence implies the local convergence in measure.) Strangely
enough, it is not mentioned by Tao (Sect. 1.5, p. 95) among 5 modes.1
Throughout, (X, S, µ) is a σ-finite measure space.
5b1 Definition. Let f, f1 , f2 , · · · : X → R be measurable functions. We say
that fn → f locally in measure, if
µ{x ∈ A : |fn (x) − f (x)| ≥ ε} → 0 as n → ∞
for all ε > 0 and all A ∈ S such that µ(A) < ∞.
This definition generalizes readily to fn : X → [−∞, +∞]; but f must
be finite a.e.
Clearly, fn → f if and only if fn − f → 0.
5b2 Exercise. It is enough to check a single sequence of sets A = A1 , A2 , . . .
such that ∪k Ak = X.
Prove it.2
If µ(X) < ∞, then it is enough to check A = X, of course; this case is
called the global convergence in measure. For µ(X) = ∞ the global version
is stronger than the local one; for example, on R, 1l[n,∞) → 0 locally but not
globally.
5b3 Remark. If fn → f and fn → g then f = g a.e. (Local convergence in
measure is meant in both cases.)
Proof (sketch): {x : |f (x) − g(x)| ≥ 2ε} ⊂ {x : |fn (x) − f (x)| ≥ ε} ∪ {x :
|fn (x) − g(x)| ≥ ε}.
1
Also not mentioned by Capiński & Kopp, nor by Jones. Mentioned by N. Lerner
“A course on integration theory”, Springer 2014 (Exercise 2.8.14 on p. 113); by F. Liese
& K.-J. Miescke “Statistical decision theory”, Springer 2008 (Def. A.11 on p. 619); and,
prominently, on Wikipedia, “Convergence
in measure”.
2
Hint: µ A \ (A1 ∪ · · · ∪ An ) → 0.
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48
5b4 Remark (sandwich). If gn ≤ fn ≤ hn a.e., gn → f and hn → f , then
fn → f . (Local convergence in measure is meant in all three cases.)
Proof (sketch): {x : |fn (x) − f (x)| ≥ ε} ⊂ {x : |gn (x) − f (x)| ≥ ε} ∪ {x :
|hn (x) − f (x)| ≥ ε}.
5b5 Exercise. If fn → 0 and gn → 0, then afn + bgn → 0 for arbitrary
a, b ∈ R. (Local convergence in measure is meant in all three cases; no
matter what is meant by ∞ − ∞.)
Prove it.
Thus, if fn → f and gn → g, then afn + bgn → af + bg (since (afn +
bgn ) − (af + bg) = a(fn − f ) + b(gn − g)).
Interestingly, the local convergence in measure is insensitive not only to
the choice of functions fn in their equivalence classes, but also to the choice
of a measure µ in its equivalence class defined as follows.
5b6 Definition. A measure ν on (X, S) is called equivalent to µ if ν = f · µ
for some f : X → (0, ∞).
This is indeed an equivalence relation; symmetry: if ν = f · µ then
µ = f1 · ν; transitivity: if ν1 = f · µ and ν2 = g · ν1 then ν2 = (f g) · µ (recall
4c26 and the paragraph after it).
If µ and ν are equivalent then they have the same null sets (think, why).
5b7 Lemma. If ν is equivalent to µ, then fn → f locally in µ if and only if
fn → f locally in ν.
Proof. Let fn → f locally in µ, ε > 0 and ν(A) < ∞; we have to prove that
ν(An ) → 0, where An = {x ∈ A : |fn (x) − f (x)| ≥ ε}. By 5b2, WLOG we
dν
and dµ
are bounded on A, since countably many such
may assume that dµ
dν
sets A can cover X (think, why). Now, µ(A)
≤ supx∈A dµ
(x) ν(A) < ∞,
dν
dν
(x) µ(An ) → 0.
thus µ(An ) → 0, and ν(An ) ≤ supx∈A dµ
5b8 Exercise. There exists a finite measure ν equivalent to µ.
Prove it.1
For such ν, local convergence in µ is equivalent to the global convergence
in ν. The latter is easy to metrize.
5b9 Exercise. If µ(X) < ∞, then the following conditions on measurable
fn : X → R are equivalent:
(a) fn → 0 in measure;
1
Hint: X = ]Ak ,
dν
dµ
= εk on Ak .
Tel Aviv University, 2015
Functions of real variables
49
R
(b) X min(1, |fn |) dµ → 0;
R |fn |
(c) X 1+|f
dµ → 0.
n|
Prove it.
We may define a metric ρ on the set1 L0 (X) of all equivalence classes of
measurable functions X → R by2
Z
min(1, |f − g|) dµ
(5b10)
ρ [f ], [g] =
X
provided that µ(X) < ∞; otherwise we use a finite equivalent measure instead.
5c
Convergence almost everywhere
As was noted in Sect. 4c, “almost everywhere” (a.e.) means “outside some
null set”. In particular, we say that fn → f a.e., if fn (x) → f (x) for all
x outside some null set. (Sometimes this convergence is called “pointwise
a.e.”.)
Once again, the definition generalizes readily to fn : X → [−∞, +∞];
but f must be finite a.e.
Once again, fn → f if and only if fn − f → 0. Also, if fn → f and
gn → g, then afn + bgn → af + bg (no matter what is meant by ∞ − ∞).
Still, (X, S, µ) is a σ-finite measure space.
5c1 Lemma. Convergence almost everywhere implies local convergence in
measure.
Proof. First, we consider monotone convergence: fn ↓ 0 a.e. In this case,
given ε > 0 and µ(A) < ∞, the sets An = {x ∈ A : fn (x) ≥ ε} satisfy An ↓ ∅,
that is, A \ An ↑ A (up to a null set), which implies µ(A \ An ) ↑ µ(A), that
is, µ(An ) ↓ 0.
Second, the general case: fn → 0 a.e. We introduce functions gn =
inf(fn , fn+1 , . . . ), hn = sup(fn , fn+1 , . . . ); they are measurable by 3c10. Almost everywhere, gn ↑ 0 (think, why), hn ↓ 0, and gn ≤ fn ≤ hn .3 By the
first part of this proof, gn → 0 and hn → 0 locally in measure. It remains to
use 5b4.
5c2 Corollary (of 5c1 and 5b3). If fn → f locally in measure and fn → g
a.e., then f = g a.e.
1
Denoted also by L0 (X, S, µ), L0 (µ), L0 (X), etc.
R
|f −g|
2
Alternatively, use X 1+|f
−g| dµ.
3
Note the zigzag sandwich!
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Functions of real variables
5c3 Example. Local convergence in measure does not imply convergence
almost everywhere.
We take the measure space [0, 1] with Lebesgue measure. For every n
there exists a partition of X = [0, 1] into n sets (just intervals, if you like)
of measure 1/n each. We combine such partitions (for odd n) into a single
infinite sequence of sets
A1 , A2 , A3 , A4 , A5 , A6 , A7 , A8 , A9 , A10 , A11 , A12 , A13 , A14 , A15 , . . .
{z
}|
{z
}
| {z } |
partition
partition
partition
such that Ak2 ] Ak2 +1 ] · · · ] A(k+1)2 −1 = X and µ(Ak2 ) = µ(Ak2 +1 ) =
1
· · · = µ(A(k+1)2 −1 ) = 2k+1
for k = 1, 2, . . . Clearly, µ(An ) → 0, therefore the
indicators fn = 1lAn converge to 0 in measure. However, lim supn fn (x) = 1
for every x, since x belongs to infinitely many An .
5c4 Exercise. If X is (finite or) countable, then local convergence in measure
implies convergence almost everywhere.
Prove it.
P
5c5 Lemma. Let µ(X) < ∞, and ρ be defined by (5b10). If n ρ(fn , fn+1 ) <
∞, then the sequence (fn )n converges a.e.
Proof. We denote gn = min(1, |fn+1 − fn |), Sn = g1 + · · · + gn ↑ S : X →
[0, ∞]. Using the Monotone Convergence Theorem 4c11,
Z
Z
Z
Z
XZ
gn dµ < ∞ .
S dµ = lim
Sn dµ = lim
g1 dµ+· · ·+ gn dµ =
X
n
X
n
X
X
n
X
P
By (4c31), S < ∞ a.e. P
For almost every x we have n min(1, |fn+1 (x) −
fn (x)|) < ∞,
(x) − fn (x)| < ∞ (think, why), therefore
P therefore n |fn+1
the series
n fn+1 (x) − fn (x) converges, that is, the sequence fn (x) n
converges.
5c6 Corollary (of 5c5 and 5c1). The metric space (L0 (X), ρ) is complete.
That is, every Cauchy sequence is converging.
It appears that convergence almost everywhere cannot be metrized (in
contrast to local convergence in measure). Moreover, no functional H :
L0 [0, 1] → [0, ∞] satisfies
H(fn ) → 0
⇐⇒
fn → 0 a.e.
Proof: otherwise we take fn such that fn → 0 in measure but not a.e.;
note that H(fn ) 6→ 0; take a subsequence gi = fni and ε > 0 such that
∀i H(gi ) ≥ ε; note that gi → 0 in measure; take a subsequence hj = gij such
that hj → 0 a.e.; then H(hj ) → 0 but H(hj ) ≥ ε, — a contradiction.
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51
5c7 Exercise.R(a) Let µ(X)
R < ∞ and ∀n |fn | ≤ 1 a.e. If fn → f locally in
measure, then X fn dµ → X f dµ. Prove it.
(b) Both assumptions (about µ(X) and |fn |) are essential; give two counterexamples.
5c8 Exercise. (a) A sequence of real numbers converges to 0 if and only if
every subsequence contains a (sub)subsequence that converges to 0;
(b) a sequence of measurable functions X → [−∞, +∞] converges to 0 locally in measure if and only if every subsequence contains a (sub)subsequence
that converges to 0 almost everywhere.
Prove it.
5c9 Exercise. If fn → f locally in measure, then ϕ ◦ fn → ϕ ◦ f locally in
measure for every continuous ϕ : [−∞, +∞] → [−∞, +∞].
Prove it.1
5d
Dominated convergence; integrals depending on a
parameter
5d1 Theorem (Dominated Convergence Theorem). Let f, f1 , f2 , · · · : X →
R and g : X → [0, ∞) be measurable functions such that
fn → f a.e. ;
∀n |fn | ≤ g a.e. ;
g is integrable .
Then f is integrable, and
Z
Z
fn dµ →
X
f dµ .
X
Proof. We did not stipulate σ-finiteness of µ, but this does not matter;
WLOG, g : X → (0, ∞), since all fn and f vanish outside {x : g(x) 6= 0};
now µ must be σ-finite, since it is equivalent to the finite measure ν = g · µ.
By (4c28),2
Z
Z
Z
Z
f
fn
dν ,
f dµ =
dν .
fn dµ =
X
X g
X
X g
It remains to use 5c7 (and 5c1).
1
2
Hint: use 5c8.
Generalized to integrable (signed) functions.
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Functions of real variables
52
5d2 Remark. The a.e. convergence (fn → f ) may be replaces with the local
convergence in measure. (Use 5c8.)
The Dominated Convergence Theorem 5d1 is useful when dealing with
integrals depending on a parameter.1 Consider a function f : R × X → R
such that its first section f (t, ·) : x 7→ f (t, x) is integrable (for every t ∈ R),
and its second sectionRf (·, x) : t 7→ f (t, x) is continuous (for every x ∈ X).
We introduce F (t) = X f (t, ·) dµ and wonder, whether F is continuous, or
not.
The function g : x 7→ supt∈R f (t, x) is measurable (since the supremum
may be taken over rational t); and if g is integrable, then F is continuous
(since tn → t implies F (tn ) → F (t) by 5d1).
Z
(5d3) If t 7→ f (t, x) is continuous and
sup |f (t, ·)| dµ < ∞ ,
X t
Z
f (t, ·) dµ is continuous.
then t 7→
X
R
R
(Warning: sup(. . . ) is generally larger than sup (. . . ).)
Moreover, it is enough if t 7→ f (t, x) is continuous at t for all x outside
some null set that may depend on t. In particular,
Z t
f dm is continuous.
(5d4)
if f : R → R is integrable, then t 7→
0
(Consider f˜(t, x) = f (x)1l[0,t] (x).) Similarly, for integrable
(and maybe,
R
d
nowhere continuous) f : R → R the function r 7→ |·|<r f dm is continuR
ous. The same holds for r 7→ |·|<r f (r, ·) dm when r 7→ f (r, x) is continuous
for all x, provided that the supremum is integrable.
When (X, S, µ) is a probability space, we get a probabilistic statement:
(5d5) If a random function is continuous almost surely (at each point separately)
and the supremum of its absolute value has finite expectation,
then its expectation is a continuous function.
5d6 Example. The integrable majorant is essential.
Consider a random function equal to 2n f (2n t) with probability 2−n , n =
1, 2, . . . ; here

1

2t − 1 for t ∈ [ 2 , 1],
f (t) = 2 − t for t ∈ [1, 2],


0
for t ∈ (−∞, 12 ] ∪ [2, ∞).
1
See Jones, Sect. 6G.
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53
Functions of real variables
Its expectation is


for t ∈ (−∞, 0] ∪ [1, ∞),
0
n
f (2 t) = 1
for t ∈ [0, 21 ],


n=1
2(1 − t) for t ∈ [ 21 , 1]
∞
X
since for t = 2−n θ, θ ∈ [ 12 , 1] we have f (2n t) = 2θ − 1 and f (2n+1
t) = 2 − 2θ;
P∞
other summands vanish. The expectation of the supremum is n=1 2n 2−n =
∞.
R
Now we wonder, whether F : t 7→ X f (t, ·) dµ is continuously differentiable. We assume that, for almost every x, f (·, x) ∈ C 1 (R); and in addition,
∂
f (t, x)| is integrable. Then, by (5d3),
that Rthe function g : x 7→ supt∈R | ∂t
∂
t 7→ X ∂t f (t, ·) dµ is continuous. But is it F 0 ? We have
f (t + h, ·) − f (t, ·) f (t + h, ·) − f (t, ·)
∂
→ f (t, ·) a.e., ≤ g(·) ;
h
∂t
h
R
R ∂
(t)
(t,·)
by 5d1, F (t+h)−F
= X f (t+h,·)−f
dµ → X ∂t
f (t, ·) dµ as h → 0. Nice;
h
h
but is f (t, ·) integrable? For now we only know that f (t + h, ·) − f (t, ·) is
integrable for all t and h.
Z
∂
1
(5d7) If t 7→ f (t, x) ∈ C (R) , and
sup f (t, ·) dµ < ∞ , and
X t∈R ∂t
f (t, ·) is integrable for some (therefore, every) t , then
Z
Z
Z
d
∂
1
t 7→
f (t, ·) dµ ∈ C (R) , and
f (t, ·) dµ =
f (t, ·) dµ .
dt X
X
X ∂t
The corresponding probabilistic statement:
(5d8) If a random function belongs to C 1 (R) almost surely,
and the supremum of absolute value of its derivative has finite expectation,
and the expectation of the function is well-defined at some (therefore, every) point,
then the derivative of the expectation is equal to the expectation of the derivative,
and is continuous.
5e
One-sided bounds
First, the monotone case.
In the Monotone Convergence Theorem 4c11, functions fn are bounded
from below by 0; equally well they may be bounded from below by an integrable function.
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54
5e1 Corollary (of 4c11). Let g, f, f1 , f2 , · · · : X → [−∞, +∞] be measurable
functions such that fn ↑ f a.e., f1 ≥ g and g is integrable. Then
Z
xZ

f dµ ∈ (−∞, +∞] .
fn dµ 
X
X
R
For the proof, just apply 4c11 to (fn − g) ↑ (f − g) and cancel
R − g.
The integrable minorant g for f1 exists if and only if X f1 dµ < ∞
(think, why).
R
R
Clearly, 5e1 still holds if ∃n X fn− dµ < ∞. Otherwise,R if ∀n X fn− dµ =
∞, 5e1 fails for two reasons. First, it may happen that X fRn dµ is ∞ − ∞
for all n, and the conclusion Ris a nonsense. Second, even if X fn dµ = −∞
for all n, it does not imply X f dµ = −∞. For a counterexample take a
nonintegrable h : X → [0, ∞), an integrable f , and consider fn = f − n1 h.
Now, the non-monotone case.
In the Dominated Convergence Theorem 5d1, functions fn are bounded
by integrable function(s) from both sides:
R −g ≤ fn ≤ g. In Example 5d6,
taking tn → 0, we get fn → 0 a.e., but X fn dµ → 1; these fn are bounded
from below (just by 0), and surely not bounded from above
(by an integrable
R
function) according to 5d1. Also, (−f
R n ) → 0 a.e., but X (−f
R n ) dµ → −1; and
(−fn ) ≤ 0. Does it mean that limn fn can deviate
R from lim
R n fn only to the
unbounded direction? Can we prove that limn X fn dµ ≤R X f dµ whenever
fn → f a.e. and ∀n fn ≤ g, g integrable,
or equivalently, X (supn fn )+ dµ <
R
∞? No, we cannot, since limn X fn dµ
R need not Rexist (try fR2 = f4 = ·· · =
f ). But we can prove that lim supn X fn dµ ≤ X f dµ = X limn fn dµ.
Moreover,
of limn fn ; we can prove that
R we do notR really need existence
lim supn X fn dµ ≤ X lim supn fn dµ whenever supn fn + is integrable.
Or we may change the signs of fn .
5e2 Proposition (Fatou’s Lemma).
Z
Z
lim inf fn dµ ≤ lim inf
fn dµ
X
n
n
X
whenever measurable functions fn : X → [−∞, +∞] are such that (inf n fn )−
is integrable.
(If fn : X → [0, ∞], then (inf n fn )− = 0 is integrable, of course.)
5e3 Remark. Let (X, S, µ) be a probability space, A1 , A2 , · · · ∈ S, An ↓
1
∅, and
R µ(An ) = n . Consider fn = n1lAn . Clearly, fn → 0 a.s., but
∀n
f dµ = 1. The distribution of (the random variable) fn (recall
X n
page 30) consists of two atoms; one atom of mass 1/n at n, the other atom
Tel Aviv University, 2015
55
Functions of real variables
of mass 1 − n1 at 0. When n tends to infinity, the first atom “escapes to
infinity”. That is, its contribution to the expectation (the integral) escapes
to ∞; its mass (probability) does not escape, it returns to (the atom at) 0.
The integrable lower bound prevents escape to −∞, but does not prevent
escape to +∞. Intuitively, Fatou’s Lemma 5e2 says that a part of the integral can escape to infinity (in the allowed direction), but cannot come from
infinity.
5e4 Remark. Nothing like that holds for lim supn fn (unless we turn to
integrable
upper bound). Indeed, according to 5c3 it can happen that fn ≥ 0,
R
f dµ → 0, but lim supn fn = 1 a.e.
X n
Proof of Prop. 5e2. We introduce1 functions gn = inf(fn , fn+1 , . . . ), note
that gn ↑ lim inf n fn , and apply Monotone Convergence Theorem 4c11 to
functions gn − g, where g = −(inf n fn )− is an integrable
lower bound of all
R
fn and therefore of all gn , too; we get (canceling g)
Z
xZ

(lim inf fn ) dµ .
gn dµ 
Thus,
Z
Z
Z
lim inf fn dµ = lim
X
n
n
X
X
gn dµ ≤ lim inf
gn dµ = lim inf
n
n
X
Z
X
n
fn dµ ,
X
since gn ≤ fn .
5e5 Exercise.
Z
Z
f dµ ≤ lim inf
X
n
fn dµ
X
whenever measurable functions f, fn : X → [−∞, +∞] are such that fn → f
locally in measure and (inf n fn )− is integrable.
Prove it.2
R
5e6 Corollary. (a) The set of all f ∈ L0 (X) such that f ≥ 0 and X f dµ ≤ 1
is closed in L0 (X) (w.r.t. the local convergence
in measure);
R
(b) the same
R holds for f such that |f | dµ ≤ 1, and more generally, for
f such that ϕ ◦ f dµ ≤ 1 for a given continuous ϕ : [−∞, +∞] → [0, ∞]
(recall 5c9).3
R
In contrast, the set of all f ∈ L0 (X) such that f ≥ 0 and X f dµ ≥ 1 is
not closed in L0 (X).
1
Similarly to the proof of 5c1.
Hint: 5c8.
3
Still more generally: for all lower semicontinuous ϕ (it means, lims→t ϕ(s) ≥ ϕ(t)).
2
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5f
Functions of real variables
56
Spaces Lp
Let ϕ : R → [0, ∞) be a convex function such that ϕ(0) = 0, ϕ(t) > 0 for all
t 6= 0, and ϕ(−t) = ϕ(t). For example, ϕ(t) = |t|p for a given p ∈ [1, ∞).
We introduce the set
Z
n
o
Bϕ = f ∈ L0 (X) :
ϕ ◦ f dµ ≤ 1 .
X
By 5e6(b), Bϕ is closed in L0 (X). Clearly, Bϕ is symmetric: (−f ) ∈ Bϕ ⇐⇒
f ∈ Bϕ . Also, Bϕ is convex:
θf + (1 − θ)g ∈ Bϕ for all f, g ∈ Bϕ and θ ∈ [0, 1] ,
R
R
R
since XR ϕ θf + (1 − θ)g dµ ≤ X θϕ(f ) + (1 − θ)ϕ(g) dµ = θ X ϕ(f ) dµ +
(1 − θ) X ϕ(g) dµ ≤ θ + (1 − θ) = 1.
In particular, rBϕ ⊂ Bϕ for r ∈ [0, 1].
We introduce the functional k · kϕ : L0 (X) → [0, ∞] by
kf kϕ = inf{r > 0 : f ∈ rBϕ } = inf{r > 0 : 1r f ∈ Bϕ } ;
the infimum is reached (since Bϕ is closed) unless the set is empty (in which
case kf kϕ = inf ∅ = +∞, of course). Convexity of Bϕ implies the triangle
inequality:
kf + gkϕ ≤ kf kϕ + kgkϕ ,
since, assuming kf kϕ + kgkϕ < ∞ (otherwise nothing to prove) we take
θ ∈ [0, 1] such that kf k = θ(kf k + kgk), kgk = (1 − θ)(kf k + kgk) and get
f ∈ θ(kf k+kgk)Bϕ , g ∈ (1−θ)(kf k+kgk)Bϕ , whence f +g ∈ (kf k+kgk)Bϕ .
Clearly, kf kϕ = 0 if and only if f = 0 a.e.
Thus, the set
Lϕ (X) = {f ∈ L0 (X) : kf kϕ < ∞}
is a vector space; nBϕ ↑ Lϕ (X) as n → ∞. Being endowed with the norm
k · kϕ it is a normed space, and moreover, a Banach space by 5f3 below.
In particular, when ϕ(t) = |t|p for a given p ∈ [1, ∞), we get the space1
Lp (X), with the norm k · kp ,
Z
1/p
kf kp =
|f |p dµ
.
X
5f1 Lemma. If fn → 0 in Lϕ (X) (that is, fn ∈ Lϕ (X) and kfn kϕ → 0),
then fn → 0 in measure (globally, and therefore locally).
1
Denoted also by Lp (X, S, µ), Lp (µ), Lp (X), etc.
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Functions of real variables
57
Proof. By convexity, t 7→ ϕ(t)/t is increasing on (0, ∞), therefore ϕ(t) ↑ ∞
as t → ∞. We have fn ∈ rn Bϕ , rn → 0. For every ε > 0,
|f (x)| ε n |f (x)|
εo
n
n
≥
=µ x:ϕ
≥ϕ
≤
µ{x : |fn (x)| ≥ ε} = µ x :
rn
rn
rn
rn
Z 1
1
1 ≤
fn dµ ≤
→ 0.
ϕ
ϕ(ε/rn ) X
rn
ϕ(ε/rn )
Let µ be σ-finite, ν a finite measure equivalent to µ, and ρ the corresponding metric on L0 (X) (introduced in (5b10)).
5f2 Exercise. (a) For every ε > 0 there exists δ > 0 such that
∀f ∈ L0 (X) kf kϕ < δ =⇒ ρ(f, 0) < ε .
(b) Every Cauchy sequence in Lϕ (X) is a Cauchy sequence in L0 (X), ρ .
Prove it.1
5f3 Proposition. The normed space Lϕ (X) is complete (that is, every
Cauchy sequence is converging).
Proof. Let f1 , f2 , · · · ∈ Lϕ (X) be a Cauchy sequence, that is, supk,l≥n kfk −
fl kϕ = εn ↓ 0. We did not stipulate σ-finiteness of µ, but this does not
matter (similarly to the proof of 5d1); WLOG, µ is σ-finite, since all fn vanish
outside the countable union ∪m,n {x : |fn | > 1/m} of sets
of finite measure.
By 5f2(b), fn are also a Cauchy sequence in L0 (X), ρ . By 5c6 there exists
f ∈ L0 (X) such that fn → f in L0 (X). We note that fn+k − fn ∈ εn Bϕ ,
and εn Bϕ is closed in L0 (X) (think, why), therefore f − fn ∈ εn Bϕ , that is,
kfn − f kϕ ≤ εn → 0.
Thus, all Lp (X) for p ∈ [1, ∞) are Banach spaces. But L2 (X) is, moreover, a Hilbert space. If f, g ∈ L2 (X), then f + g ∈ L2 (X), thus, f 2 , g 2 , (f +
g)2 and 2f g = (f + g)2 − f 2 − g 2 are integrable. We introduce the inner
product
Z
hf, gi =
f g dµ for f, g ∈ L2 (X) ;
X
it is bilinear, symmetric, and hf, f i = kf k22 , which shows that L2 (X) is a
Hilbert space. Every finite-dimensional subspace of L2 (X) is a Euclidean
space, therefore, linearly isometric to Rn with the Euclidean norm |x| =
1
Hint: (a) use 5f1; (b) use (a).
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Functions of real variables
58
(x21 + · · · + x2n )1/2 . In particular, every 2-dimensional subspace is a Euclidean
plane. Applying this fact to the subspace spanned by f, g we get hf, gi =
kf k2 kgk2 cos θ for some θ (the angle between the vectors f, g), and therefore
−kf k2 kgk2 ≤ hf, gi ≤ kf k2 kgk2
(Schwarz inequality).
Back to Lp (X).
5f4 Lemma. If µ(X) < ∞ and 1 ≤ p ≤ q < ∞, then Lp (X) ⊃ Lq (X) and
sup kf kp < ∞ .
kf kq =1
R
Proof. 1 If kf kq = 1, then X |fR|q dµ = 1; we note that
|f |p ≤ max(1, |f |q ) ≤
R
1 + |f |q (think, why), and get X |f |p dµ ≤ µ(X) + X |f |q dµ = µ(X) + 1,
that is, kf kp ≤ µ(X) + 1 1/p .
5f5 Remark. (a) On (0, 1) (with Lebesgue measure), the function t 7→ t−α
belongs to Lp if and only if p < 1/α (check it), which shows that the inclusion
in 5f4 is generally strict: Lp (X) % Lq (X).
(b) On (1, ∞) (with Lebesgue measure), the function t 7→ t−α belongs to
Lp if and only if p > 1/α (check it), which shows that the inclusion in 5f4
may fail when µ(X) = ∞.
5f6 Lemma. If X is countable and inf x∈X µ({x}) > 0, then Lp (X) ⊂ Lq (X)
for 1 ≤ p ≤ q < ∞, and
sup kf kq < ∞ .
kf kp =1
R
P
p
p
Proof. If kf kp = 1, then
x∈X |f (x)| µ({x}) = X |f | dµ = 1. Introducing ε = inf x∈X µ({x}) >R 0 we note that
|f (x)|
≤ 1ε 1/p , therefore
R
|f |q ≤ 1ε (q−p)/p |f |p , and get X |f |q dµ ≤ 1ε (q−p)/p X |f |p dµ = 1ε (q−p)/p ,
1 1
that is, kf kq ≤ 1ε p − q .
5f7 Exercise. (a) lim inf p→∞ kf kp ≥ kf k∞ , where
kf k∞ = ess supx∈X |f (x)| = inf{t ∈ (0, ∞) : |f (·)| ≤ t a.e. }
(the infimum is reached, unless the set is empty, in which case kf k∞ = inf ∅ =
+∞, of course).
(b) If µ(X) < ∞ then kf kp → kf k∞ as p → ∞.
1
See Capiński & Kopp, Th. 5.25. In fact, the supremum is reached on constant f and
1
1
is therefore equal to (µ(X)) p − q .
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59
(c) The set
L∞ (X) = {f ∈ L0 (X) : kf k∞ < ∞} ,
endowed with k · k∞ , is a Banach space.
Prove it.1 Try to extend 5f4 and 5f6 to q = ∞.
Index
Banach space, 57
Fatou’s Lemma, 54
complete (space), 50
convergence
almost everywhere, 46, 49
in measure (local), 47
in measure (global), 47
pointwise almost everywhere, 49
uniform almost everywhere, 46
Hilbert space, 57
Dominated Convergence Theorem, 51
equivalent measures, 48
1
Hint: (a) kf kp ≥ c µ{x : |f (x)| ≥ c}
convergence outside a null set.
Bϕ , 56
L0 , 49
L∞ , 59
Lϕ , 56
Lp , 56
|| · ||∞ , 59
|| · ||ϕ , 56
|| · ||p , 56
ρ, 49
1/p
; (b) use (a); (c) 5c6 aside, use uniform